4.3: Propositions, Inferences, and Judgments

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20 Propositions, Inferences, and Judgments^³⁵

Consider the following two statements:

- She is annoying, but I love her. - I love her, but she is annoying.

What is being said here? Technically speaking, “but” functions as a conjunction and the two sentences mean logically the exact same thing. They mean, “I love her and she is annoying” and nothing more. It is difficult to capture the subtleties of language in logical systems sometimes, and this is one example. In the first case, the emphasis is on love, and in the second case, it is on annoyance. Today’s Lesson is about some of the subtleties of language and how we understand logic on a day-to-day basis in our ordinary language. While the proposition (a statement that is subject to being True or False) says one thing technically, its regular-language meaning is significantly different. Judgments are when we determine whether the statement is actually True or False. So determining the Truth of the above statement would be one thing in Logic, and another in real-life. Why do we care about the Truth or Falsity of statements? Because knowing whether they are True or False allows us to figure out what else we know. Regardless of their being True or False, there are inferences (statements that necessarily follow from assuming certain statements are True) that we can understand. The process is generally the same: given what we’ve been told, what else can we infer? Logical equivalency means that two statements are Truth-functionally equivalent, which means that they are both True under all the same circumstances. Essentially, this means that whenever one of them is True, the other is True as well. If we can understand which statements are logically equivalent, then we can understand the basic idea of entailments. Single statements can give us some inferences like equivalency, and this is where we will begin before moving to more complex inferences in the forms of proofs and deductions. For now, we’re going to get into some more detail on disjunctions in order to understand what we can infer an understand from basic statements.

Exclusive vs. Nonexclusive Disjunctions^³⁶

NOTE: ∙ = & in the text below. There are different notations and in some of them they use ∙ to represent conjunctions.

The connectives “or” and “either … or” are used in two distinct ways in daily discourses. When a host asks you “Coffee or Tea?”, it is implicitly implied that you should choose coffee or tea, but not both. The connective “or” is used in the exclusive sense to mean “one or the other, but not both.” Afterwards, when the host asks you again “Cream or sugar?”, you can respond by saying “Both, please.” Now the connective is used in the nonexclusive sense of “one or the other, or both.”

Here is an example of “either … or …” used in the nonexclusive sense:

Either fire or smoke can damage the paintings.

F ∨ S

F: Fire can damage the paintings.

S: Smoke can damage the paintings.

If either fire or smoke alone can damage the paintings, then the two together can damage the paintings.

In Propositional Logic, the wedge “∨” is used to symbolize nonexclusive disjunctions. So the sentence is symbolized as F ∨ S.

By contrast, in the next sentence “either … or …” is used in the exclusive sense.

The Federal Reserve will either raise interest rates or leave them intact.

∼(R ∙ L)

R: The Federal Reserve will raise interest rates.

L: The Federal Reserve will leave interest rates intact.

∼(R ∙ L). The first conjunct R ∨ L means that the Federal Reserve will do one or the other, or both. But the second conjunct ∼(R ∙ L) says that the Federal Reserve will not do both. So together, they capture the meaning of “one or the other, but not both.”

How to Symbolize “unless”

A compound sentence formed with the connective “unless” can be symbolized as a conditional or a biconditional, depending on the meaning of the sentence. It can also be symbolized as a disjunction. But in doing so, we need to pay attention to whether it is the exclusive or the nonexclusive disjunction. The sentence

Jeff cannot graduate unless he completes all the GE requirements.

We can also rewrite this as

Either Jeff completes all the GE requirements or he cannot graduate.

∼G ∼G, because it is possible that Jeff completes all the GE requirements but still cannot graduate due to, say, having not yet met the total unit requirement. ∼G is logically equivalent to ∼G ∨ C. As a result, we can symbolize it as ∼G ∨ C.

Jeff cannot graduate unless he completes all the GE requirements.

∼G ∨ C

“Not … both …” and “Both … not …”

It is important not to conflate “Not … both …” and “Both … not …”. Compare these two sentences:

Not both Monet and Chopin are painters.

It is not the case that Monet is a painter and Chopin is a painter.

∼(M ∙ C)

M: Monet is a painter.

C: Chopin is a painter.

Both Dvořák and Schubert are not painters.

Dvořák is not a painter and Schubert is not a painter.

∼S

D: Dvořák is a painter.

S: Schubert is a painter.

The first sentence denies that both Monet and Chopin are painters. That is, it says that at least one of them is not a painter. It can be rephrased as

Either Monet or Chopin is not a painter.

Either Monet is not a painter or Chopin is not a painter.

∼C ∼C if we expand it fully as

Either Monet is not a painter or Chopin is not a painter.

∼C.

By contrast, the second sentence is a conjunction.

Both Dvořák and Schubert are not painters.

Dvořák is not a painter and Schubert is not a painter.

∼S

D: Dvořák is a painter.

S: Schubert is a painter.

Moreover, this is logically equivalent to both sentences below:

Not either Dvořák or Schubert is a painter.

It is not the case that either Dvořák is a painter or Schubert is a painter.

∼(D ∨ S)

Neither Dvořák nor Schubert is a painter.

It is not the case that either Dvořák is a painter or Schubert is a painter.

∼(D ∨ S)

∼S is logically equivalent to ∼(D ∨ S).

The following formulas sum up the differences between “Not … both …” and “Both … not …” and show how to symbolize them:

∼q ∼q = Not either p or q = ∼(p ∨ q) = Neither p nor q = ∼(p ∨ q)